Cycling is where people continuously move in and move out villagers. Some do it in hopes of getting dreamies, some do it for money, and some just do it to help people get their dreamies (or a combination of the three!). In boxes means that the villager is ready to move, and that someone can take them to their town. As for getting Henry, it's up to luck. If you find Henry, not many people will want him (as he's not too popular), but he is more likely to just be autovoided (where no one gets the villager, they just leave) since he is a tier five villager. Basically, it depends. I recommend checking cycling threads to see if he's in the town, and if he is in a cycling town, watch the thread and wait for him to go in boxes so you can get him.
I hope that answered your question!
I hope that answered your question!